Finite Group Databasebeta

← #136033

Subgroups — #136033

Order 512.

525 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14168C211yesmaximal, characteristic
2C2×#1324C211yesmaximal, characteristic
3#11211C211yesmaximal, characteristic
4#8856C212yesmaximal
5#7251C212yesmaximal
6#2316C2211yescharacteristic
7C2×#382C2211yescharacteristic
8C2×#376—21no—
9#1852—22no—
10C2×#343C2211yescharacteristic
11#1736—21no—
12C2×#339—21no—
13C2×#324C2212yes—
14#1441—21no—
15#1441C2212yes—
16#1436—22no—
17#1436—22no—
18#1324C414yescomplemented
19#1294—22no—
20C22×M16—21no—
21C2×#117—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4D411yescharacteristic
24C22×C4⋊C4D412yes—
25C22×C4⋊C4C2311yescharacteristic
26C22×C42—22no—
27C22×C42D411yescharacteristic
28C2×#107—42no—
29C2×C22⋊C8—22no—
30#389—42no—
31#382—22no—
32#382C2×C412yes—
33#382Q812yes—
34#380—22no—
35#376—42no—
36#376—42no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95D412yes—
45#343—22no—
46#340—42no—
47#340—42no—
48#339—42no—
49#324—22no—
50#324C2×C414yes—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4C2×D411yescharacteristic
56C23×C4C2×D412yes—
57C2×M16—22no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C22×C8—41no—
64C22×C8—42no—
65#117—41no—
66#117—41no—
67#117—41no—
68#117—41no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—81no—
81C2×C4⋊C4C4⋊C412yes—
82C2×C4⋊C4C22×C411yescharacteristic
83C2×C4⋊C4C2×Q811yescharacteristic
84C2×C4⋊C4#4112yes—
85C2×C22⋊C4—41no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—44no—
90C2×C42—44no—
91C2×C42—44no—
92C2×C42—41no—
93C2×C42C4⋊C412yes—
94C2×C42D812yes—
95C2×C42Q1612yes—
96#107—84no—
97C22⋊C8—24no—
98C22⋊C8—42no—
99#95—24no—
100#95—24no—
101#95—42no—
102#95—44no—
103#95—44no—
104#95—42no—
105#95—42no—
106#95—42no—
107#95—42no—
108#95—42no—
109#95—81no—
110C24#12011yescharacteristic
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4C2×C4⋊C411yescharacteristic
145C22×C4C4×D412yes—
146C22×C4C22⋊Q812yes—
147C22×C4#12311yescharacteristic
148C22×C4C2×D811yescharacteristic
149C22×C4C2×Q1611yescharacteristic
150C22×C4AGL(1,ℤ/8)11yescharacteristic
151C22×C4#13711yescharacteristic
152M16—42no—
153M16—42no—
154M16—81no—
155M16—81no—
156C2×C8—44no—
157C2×C8—42no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8—82no—
161C2×C8—82no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—81no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C22⋊C4—81no—
175C22⋊C4—81no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—416no—
180C42—81no—
181C42—81no—
182C42#10714yescomplemented
183C23—22no—
184C23—22no—
185C23#38511yescharacteristic
186C23#44711yescharacteristic
187C23#45111yescharacteristic
188C23#45411yescharacteristic
189C23#45611yescharacteristic
190C23B(4,𝔽2)11yescharacteristic
191C23#45811yescharacteristic
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4C2×#10711yescharacteristic
239C2×C4#42811yescharacteristic
240C2×C4#48011yescharacteristic
241C2×C4#48411yescharacteristic
242C8—82no—
243C8—84no—
244C22—22no—
245C22—22no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—22no—
252C22#159811yescharacteristic
253C22#160511yescharacteristic
254C22#161011yescharacteristic
255C22#161711yescharacteristic
256C22#185211yescharacteristic
257C22#185711yescharacteristic
258C22#186711yescharacteristic
259C22#219712yes—
260C22#220412yes—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—44no—
266C4—42no—
267C4—42no—
268C4—84no—
269C4—84no—
270C4—82no—
271C4—162no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2#885611yescharacteristic
277C2#1215412yes—
278C2#1225812yes—
279C2#1296511yescharacteristic
280C2#1297211yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
18. #1324C4semidirect
182. C42#107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#141682
C2×#13242
#112112
#88562
#72512